### 数学代写|交换代数代写commutative algebra代考|MATH3303

statistics-lab™ 为您的留学生涯保驾护航 在代写交换代数commutative algebra方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写交换代数commutative algebra代写方面经验极为丰富，各种代写交换代数commutative algebra相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|交换代数代写commutative algebra代考|Vector Bundles on a Smooth Compact Manifold

Here, we give some motivations for finitely generated projective modules and localization by explaining the example of vector bundles on a compact smooth manifold. Two important particular cases are tangent and cotangent bundles corresponding to $C^{\infty}$ vector fields and to $C^{\infty}$ differential forms.
We will use the term “smooth” as a synonym for “of class $C^{\infty} .$,”
We will see that the fact that the sphere cannot be combed admits a purely algebraic interpretation.

In this section, we consider a smooth real differentiable manifold $V$ and we denote by $\mathbf{A}=C^{\infty}(V)$ the real algebra of global smooth functions on the manifold.
Some Localizations of the Algebra of Continuous Functions
Let us first consider an element $f \in \mathbf{A}$ along with the open set (open subset of the manifold $V$ to be precise)
$$U={x \in V \mid f(x) \neq 0}$$
and let us see how we can interpret the algebra $\mathbf{A}[1 / f]$ : two elements $g / f^{k}$ and $h / f^{k}$ are equal in $\mathbf{A}[1 / f]$ if and only if for some exponent $\ell$ we have $g f^{\ell}=h f^{\ell}$ which means precisely $\left.g\right|{U}=\left.h\right|{U}$.

It follows that we can interpret $\mathbf{A}[1 / f]$ as a sub-algebra of the algebra of smooth functions on $U$ : this sub-algebra has as elements the functions which can be written as $\left(\left.g\right|{U}\right) /\left(\left.f\right|{U}\right)^{k}$ (for a given exponent $k$ ) with $g \in \mathbf{A}$, which a priori introduces certain restrictions on the behavior of the function on the border of $U$.

To avoid having to deal with this difficult problem, we use the following lemma.

## 数学代写|交换代数代写commutative algebra代考|Vector Bundles and Finitely Generated Projective Modules

Now recall the notion of a vector bundle over $V$.
A vector bundle is given by a smooth manifold $W$, a smooth surjective mapping $\pi$ : $W \rightarrow V$, and a structure of a finite dimensional vector space on every fiber $\pi^{-1}(p)$. In addition, locally, all this must be diffeomorphic to the following simple situation, called trivial:
$$\pi_{1}:\left(U \times \mathbb{R}^{m}\right) \rightarrow U,(p, v) \mapsto p,$$
with $m$ that can depend on $U$ if $V$ is not connected. This means that the structure of the (finite dimensional) vector space on the fiber over $p$ must “properly” depend on $p$.

Such an open set (or subset) $U$, which trivializes the bundle, is called a distinguished open set (or subset).

A section of the vector bundle $\pi: W \rightarrow V$ is by definition a mapping $\sigma: V \rightarrow W$ such that $\pi \circ \sigma=\operatorname{Id}_{V}$. We will denote by $\Gamma(W)$ the set of smooth sections of this bundle. It is equipped with a natural $\mathbf{A}$-module structure.

Now suppose that the manifold $V$ is compact. As the bundle is locally trivial there exists a finite covering of $V$ by distinguished open subsets $U_{i}$ and a partition of the unity $\left(f_{i}\right){i \in \llbracket 1 . s \rrbracket} \rrbracket$ subordinate to the open cover $U{i}$ : the support of $f_{i}$ is a compact set $K_{i}$ contained in $U_{i}$.

We notice from I.emma $1.1$ that the algebras $\mathbf{A}\left[1 / f_{i}\right]=C^{\infty}(V)\left[1 / f_{i}\right]$ and $C^{\infty}\left(U_{i}\right)\left[1 / f_{i}\right]$ are naturally isomorphic.

If we localize the ring $\mathbf{A}$ and the module $M=\Gamma(W)$ by making $f_{i}$ invertible, we obtain the ring $\mathbf{A}{i}=\mathbf{A}\left[1 / f{i}\right]$ and the module $M_{i}$. Let $W_{i}=\pi^{-1}\left(U_{i}\right)$. Then, $W_{i} \rightarrow U_{i}$ is “isomorphic” to $\mathbb{R}^{m_{i}} \times U_{i} \rightarrow U_{i}$. Thus it boils down to taking a section of the bundle $W_{i}$, or to taking the $m_{i}$ functions $U_{i} \rightarrow \mathbb{R}$ which make a section of the bundle $\mathbb{R}^{m_{i}} \times U_{i} \rightarrow U_{i}$. In other words, the module of the sections of $W_{i}$ is free and of rank $m$.

Since a module that becomes free after localization in a finite number of comaximal elements is finitely generated projective (local-global principle V-2.4), we then get the direct part (point 1 ) of the following theorem.

## 数学代写|交换代数代写commutative algebra代考|Vector Bundles on a Smooth Compact Manifold

$$U=x \in V \mid f(x) \neq 0$$

## 数学代写|交换代数代写commutative algebra代考|Vector Bundles and Finitely Generated Projective Modules

$$\pi_{1}:\left(U \times \mathbb{R}^{m}\right) \rightarrow U,(p, v) \mapsto p,$$

$W_{i}=\pi^{-1}\left(U_{i}\right)$. 然后， $W_{i} \rightarrow U_{i}$ 是“同构的” $\mathbb{R}^{m_{i}} \times U_{i} \rightarrow U_{i}$. 因此，它归结为获取捆绑包的一部分 $W_{i}$ ，或采 取 $m_{i}$ 功能 $U_{i} \rightarrow \mathbb{R}$ 构成捆绑包的一部分 $\mathbb{R}^{m_{i}} \times U_{i} \rightarrow U_{i}$. 换句话说，模块的部分 $W_{i}$ 是自由的和有等级的 $m$.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。